In my Complex Analysis course the following problem was given after teaching "Zeros of analytic function are isolated" and the Identity Theorem. So I was supposed to solve the problem using above theorems.But I could not manage to solve it using these theorems.Please help me to solve the problem. Thnx in advance.
Prove that $\sin (z+w)= \sin z\cos w +\cos z \sin w \forall z,w \in \mathbb C$
What I done so far:
Fixing $w\in \mathbb R$ and considering $f(z)=\sin(z+w),g(z)=\sin z\cos w +\cos z \sin w$ I get that $f$ and $g$ agree on $\mathbb R$ hence they agree on $\mathbb C$. And this is true $\forall w\in \mathbb R$. But I can not figure out how to solve when $w\in \mathbb C$?
I know the problem can be solved just putting values of $\sin,\cos$ in terms of exponential function but I want to solve using the theorems I have learned.
May be I am missing something very easy,I appologise for that.
Please help me. Thnx again.